More on the Geometry of Mκn

  • Martin R. Bridson
  • André Haefliger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 319)


In this chapter we return to the study of the model spaces M n k . We begin by describing alternative constructions of ℍn = M −1 n attributed to Klein and Poincaré9. In each case we describe the metric, geodesics, hyperplanes and isometries explicitly. In the case of the Poincaré model, this leads us naturally to a discussion of the Möbius group of S n and of the one point compactification of En. We also give an explicit description of how one passes between the various models of hyperbolic space. In the final paragraph we explain how the metric on M k n can be derived from a Riemannian metric and give explicit formulae for the Riemannian metric.


Hyperbolic Space Geodesic Segment Cross Ratio Geodesic Line Hyperbolic Distance 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin R. Bridson
    • 1
  • André Haefliger
    • 2
  1. 1.Mathematical InstituteUniversity of OxfodOxfordGreat Britain
  2. 2.Section de MathématiquesUniversité de GenèveGenève 24Switzerland

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